Answer: [tex]\text{P\lparen X \le4\rparen is 0.8263}[/tex]Explanation:
Given:
n = 8, p = 0.4
To find:
P(X ≤ 4)
To determine the given probability, we will apply the binomial probability formula:
[tex]\begin{gathered} $P(x=X)=^nC_x\times p^x\times q^{n-x}$ \\ n\text{ = number of trials} \\ p\text{ = number of success} \\ q\text{ = number of failure} \end{gathered}[/tex][tex]P\left(X≤4\right)\text{ = P\lparen x = 0\rparen + P\lparen x = 1\rparen + P\lparen x =2 \rparen + P\lparen x = 3\rparen + P\lparen x = 4\rparen}[/tex][tex]\begin{gathered} p\text{ + q = 1} \\ q\text{ = 1- p} \\ q\text{ = 1 - 0.4} \\ q\text{ = 0.6} \\ \\ P(x=0)=\text{ }^8C_0\times(0.4)^0\times(0.6)^{8-0} \\ P(x\text{ = 0\rparen = 0.01679616} \\ \\ P(x=1)=\text{ }^8C_1\times(0.4)^1\times(0.6)^{8-1} \\ P(x\text{ = 1\rparen = 0.08957952} \end{gathered}[/tex][tex]\begin{gathered} P(x=2)=\text{ }^8C_2\times(0.4)^2\times(0.6)^{8-2} \\ P(x\text{ = 2\rparen = 0.20901888} \\ \\ P(x=3)=\text{ }^8C_3\times(0.4)^3\times(0.6)^{8-3} \\ P(x\text{ = 3\rparen = 0.27869184} \\ \\ P(x=4)=\text{ }^8C_4\times(0.4)^4\times(0.6)^{8-4} \\ P(x\text{ = 4\rparen = 0.2322432} \end{gathered}[/tex][tex]\begin{gathered} P(X≤4)\text{ = 0.01679616 + 0.08957952 + 0.20901888 + 0.27869184 + 0.2322432} \\ \\ P\left(X≤4\right)\text{ = 0.8263296} \\ \\ To\text{ 4 decimal place, P\lparen X \le4\rparen is 0.8263} \end{gathered}[/tex]
